Found a solution.
After reading an Ouya review today, the reviewer was all gaga about using AirDroid to manage your Ouya. I had played with this class of "web desktop" apps years ago with my first Android phone. You run the app and then connect to your Android device with your browser. The app gives you a desktop that lets you manipulate your device.
Unfortunately AirDroid is a paid app and costs $1.99 a month. The free version is limited to 100MB of transfers per month which just isn't enough if you're transferring game images for your Ouya emulators.
So instead I found 3CX DroidDesktop which does what I need to do. Here it is on Google Play:
DroidDesktop on Google Play
But of course we can't use Google Play on our Ouya so here is where you can download an APK:
DroidDesktop at android-appstore
Now I thought there used to be a sideloading menu on the Ouya. I guess the last update did away with it. But fortunately the guy who did Mupen64PlusAE (one of the emulators) released FilePwn:
FilePwn on OuyaForum
Which lets you surf the Ouya file system and install packages. Install it through the normal Ouya method. Looks like it's free (as in beer free.)
DroidDesktop isn't a perfect solution. You have to specifically run it when you want to perform file management and the way the Ouya is setup an app takes over the machine and I have no idea how to switch away and do the multitasking I know Android is capable of. But it works and does *not* require rooting. I was able to surf to my USB key and copy and delete files from a Windows workstation.
Edit:
Whoops. FilePwn was *not* released by PaulsCode.com but by littleguy77. Paul just gave him a thumbs-up. My bad.